Portfolio theory is summarized by a tradeoff between mean returns and risk. In other words, it models an optimal portfolio as one guided by risk aversion. It can be turned into an explicit model of investor behaviour as a constrained optimization problem.
Begin with a finite number of assets that the investor can invest in, resulting in realized returns $R_i$, $i=1,2,\dots,N$. When investing returns are unrealized and are a random variable. If we model the investor as maximizing expected utility then in general we need to know the complete joint distribution of all the returns. On the other hand, a simpler model tracks only the mean returns, $E[R_i]$, and the variance matrix of returns $\Omega = Var[R]$, where $\Omega$ is the symmetric $N\times N$ matrix of variances and covariances of returns.
The investor must allocate their wealth over assets by choosing a weight $w_i$, the fraction of their total wealth invested in asset $i$. The weights sum to $1$ but if shorting an asset is allowed some weights can be negative. Let $w$ be the vector of weights. The objective of the investor is to maximize an objective that is a weighted average of returns and risk:
$${\max}_{w : \iota w = 1}\ q w^{\,\prime} E[R] - (1-q) w^{\,\prime}\Omega w.\nonumber$$
For $q=1$ all that matters is expected return. For $q=0$ all that matters is low risk.
A related problem is to minimize risk subject to a required expected return of the portfolio.